Munich Personal RePEc Archive

Risk prevention of land flood: A

cooperative game theory approach.

Álvarez, Xana and Gómez-Rúa, María and Vidal-Puga, Juan

Universidade de Vigo, Universidade de Vigo, Universidade de Vigo

2019

Online at https://mpra.ub.uni-muenchen.de/91515/

MPRA Paper No. 91515, posted 29 Jan 2019 16:16 UTC

Risk prevention of land flood: A cooperative game

theory approach∗

Alvarez, Xana † Gomez-Rua, Marıa ‡ Vidal-Puga, Juan §

January 2, 2019

Abstract

Protection against flood risks becomes increasingly difficult for economic and

hydrological reasons. Therefore, it is necessary to improve water retention through-

out catchment with a more comprehensive approach. Strategies in the land use and

measures that are designed to prevent flood risks involve land owners. So, justice

issues appear. This paper studies the application of game theory through a coop-

erative game in order to contribute the resolution of possible agreements among

owners and to establish cost / benefit criteria. It is a methodological contribution

where land use management for flood retention is analyzed. Specifically, we concen-

trate on enhancing upstream water retention focusing on the role that forests have

as natural water retention measures. This study shows a framework for allocating

the compensations among participants based on cooperative game theory and tak-

ing into account a principle of stability. We show that it is possible to establish

distribution rules that encourage stable payments among land owners. This contri-

bution shows the suitability of this method as a flood risk management tool and as

a guide to help decision-making. Compensations and benefits could be established

to raise awareness and encourage land owners to cooperate.

∗The authors acknowledge funding and networking support by the COST Action CA16209: Natural

Flood Retention on Private Land (LAND4FLOOD). Financial support by the Spanish Ministerio de

Economıa y Competitividad (ECO2014-52616-R), Ministerio de Economıa, Industria y Competitividad

(ECO2017-82241-R), and Xunta de Galicia (GRC 2015/014) is also gratefully acknowledged.†Escola de Enxenarıa Forestal. Universidade de Vigo. Campus A Xunqueira. 36005 Pontevedra.

Spain. E-mail: xaalvarez@uvigo.es‡Corresponding author. Facultade de Ciencias Economicas e Empresariais. Universidade de Vigo.

Campus Lagoas-Marcosende, 36310 Vigo. Spain. Phone: +34986813506. E-mail: mariarua@uvigo.es.§Facultade de Ciencias Sociais e da Comunicacion. Universidade de Vigo. Campus A Xunqueira.

36005 Pontevedra. Spain. E-mail: vidalpuga@uvigo.es

1

Keywords: game theory, land management, flood mitigation, land use, com-

pensations, decision-making,

1 Introduction

The economic and social development, in the absence of adequate territory and natural

resources planning and management, has triggered in different environmental problems.

One of the most significant hazards are flood events (Doocy et al., 2013). Floods endanger

lives and cause human tragedy as well as heavy economic losses. A study conducted by

the European Environment Agency (2011) stated that Europe suffered over 213 major

damaging floods between 1998 and 2009, having caused 1,126 deaths, the eviction of about

half a million people and at least EUR 52 billion in insured economic losses. According

to the latest studies, everything seems to indicate that a higher flood risk and greater

economic damage in Europe will happen in the near future (Jongman et al., 2015; Cook,

2017).

There are different factors that lead to damaging floods. Mainly, damages have been

attributed to increasing exposure due to high population growth and economic develop-

ment in areas prone to floods (Bouwer, 2011; Neumayer and Barthel, 2011; Field et al.,

2012; Visser et al., 2014). On the one hand, there is the occupation of the territory by

the population attending only to criteria of availability and access to resources. In this

case, population growth is increasing the likelihood of the overuse of land in flood-prone

areas (Larsen, 2009). In addition to this, it must also be considered that most cities

are located on these zones. All of these indicate a mistaken territory planning. Risk to

human life and property increases considerably in these potentially flooded areas. On

the other hand, these are suitable for agriculture (Cobourn and Lewis, 2011) with fertile

lands and close to fluvial channels for irrigation. There are also artificial infrastructures

that alter natural dynamics of a river and, therefore, the fluvial system (Nilsson and

Berggren, 2000; Lehner et al., 2011). Some examples are fish farms, reservoirs, canals,

etc. For that reason, those infrastructures that imply water storage must be managed

according to flood risk criteria (Plate, 2002). Another factor that affects flood risks is

the potential of floodplains and the adjacent land to the rivers for a land use change.

Consequently, the land use change affects the hydrology that determines flood hazard

(Wheater and Evans, 2009). And finally, climate change also arises as one of the most re-

cent factors that has worsened the incidence of floods (Milly et al., 2002; Brouwer et al.,

2007; Wilby and Keenan, 2012; Hirabayashi et al., 2013; van der Pol et al., 2017). A

2

study conducted by Alfieri et al. (2015) concluded that the socio-economic impact of

river floods in Europe would increase by an average 220% due to climate change by the

end of the 21st century. Floods are natural phenomena but climate change can influence

rainfall patterns and intensities (Kundzewicz et al., 2014) and, consequently, this could

influence the flood hazard (Milly et al., 2002; Rojas et al., 2013; Dankers et al., 2014).

In the last decades, floods are being (1) evaluated with the aim to identify the main

reasons why they occur, (2) mitigated in order to minimize their frequency and risk, and

(3) reduced and limit their impacts with the right measures. The last group corresponds

with the protective measures. Traditionally, these measures are based on the so-called

grey infrastructure, such as dikes, dams, and other concrete structures (Rasid and Paul,

1987; Roth and Winnubst, 2014; Balica et al., 2015). However, due to increase of land

use by human populations, this grey infrastructure may be not sufficient by itself to cope

with dynamic flood risk (Tempels and Hartmann, 2014; Nquot and Kulatunga, 2014;

Mustafa et al., 2018). A promising alternative is the use of nature-based solutions such

as the so-called Natural Water Retention Measures (NWRM) as a complement to grey

infrastructure (Zelenakova et al., 2017; Brody et al., 2017; Bhattacharjee and Behera,

2017, 2018). The challenge is to consider multifunctional land uses. They have the

potential to enable temporary flood retention and storage, stimulating the provision of

other ecosystem services.

Since the NWRM are usually and primarily implemented on private land, a com-

promise between flood risk management and land exploration is needed (Scherer, 1990;

Hartmann, 2016; Thaler et al., 2016). Flood management through an integrated ap-

proach combining structural and land use planning measures (Rezende, 2010; Barbedo

et al., 2014) is an efficient method of reducing flooding (Miguez et al., 2012). According

to different experts (e.g., Directorate-General for Environment (European Commission)

(2016); Machac et al. (2018)), politics such as Directive 2007/60/EC and the “Blueprint

to Safeguard Europe’s Water Resources” (European Commission, 2007), as well as The

Working Group F on Floods (2012), there are two main options for flood protection:

to control and retain floods upstream and try to adapt land uses downstream. The last

alternative has been widely analysed (e.g., Temmerman et al. (2013); Aerts et al. (2014)),

mainly because of the urgency of protecting the safety of people. There are large-scale

modelling of flood hazard (Milly et al., 2002; Pappenberger et al., 2012; Dankers et al.,

2014) and smaller scale too (te Linde et al., 2011; de Moel et al., 2015). However, in the

downstream area we can find the largest water volumes, and its topography, normally

flat, does not help the drain. For these reasons, one of the most effective flood protec-

3

tion measures is to provide more capacity for water retention and flows regulation in the

headwaters of river basins. This is the focus that will be analysed in this study.

Specifically, we explore floods reduction through actions carried out upstream. The

question that arises is the following: what can make the land owners voluntarily decide to

change the uses of their lands in order to reduce the flooding risk? The main challenge is

to reach the best agreements upstream-downstream (Machac et al., 2018). With this aim,

game theory is selected as a negotiation tool in this contribution. In particular, we need

to take into consideration multiple aspects such as economic issues (for example, how to

compensate for or incentive flood retention services), property rights (e.g., how to allow

temporary flood storage on private land), public participation (e.g., how to ensure the

involvement of private landowner), and issues of public subsidies (e.g., how to integrate

flood retention in agricultural subsidies).

Among these different aspects, in this paper we focus on a key question: How can

land owners be encouraged (or compensated) to adapt their land use and its management

strategies in a way that allows for an increase in their water retention capacity? In order

to do so, we apply cooperative game theory. This mathematical tool, first developed

by a seminal book by von Neumann and Morgenstern (1944), allows to analyze and

solve allocation situations where two or more agents (or players) have different interests.

As opposed to decision theory, where these interests are unique or coincide, and zero-

sum games, where these interests are incompatible, cooperative game theory focuses on

situations where a mutually beneficial compromise is possible. Moreover, it differs from

non-cooperative game theory in that the allocation can be done from a centralized point

of view, instead of a non-cooperative bargaining among the players.

Cooperative game theory has been applied to many areas, such as economics (Shapley

and Shubik, 1974), social sciences (Myerson, 1992), political science (Baron and Ferejohn,

1989), optimization (Curiel, 2011), health (Gonzalez and Herrero, 2004) and environmen-

tal management (Lin and Li, 2016). In particular, cooperative game theory has been used

in water resources management. Parrachino et al. (2006b); Zara et al. (2006); Parrachino

et al. (2006a) provide the basics as well as a review of some applications of cooperative

game theory to issues of water resources.

Other applications include acid rain pollution (Kaitala and Pohjola, 1998), water

resource system models (Lund and Palmer, 1997), water allocation (Wang et al., 2003),

groundwater conflicts (Raquel et al., 2016), transboundary river basins (Gengenbac et al.,

2010; Alcalde-Unzu et al., 2015; Shi et al., 2016; Li et al., 2018), and negotiation of marine

spatial allocation agreements (Kyriazi et al., 2017). Non-cooperative game theory has also

4

been applied to water management problems (Bogardi and Szidarovszky, 1976; Carraro

et al., 2007; Madani, 2010; Lee, 2012) and water right conflicts (Bergantinos and Lorenzo,

2004; Zanjanian et al., 2018).

Other research applies game theory to natural disaster management (see Seaberg

et al. (2017) for a recent survey), but very few specifically devoted to flood risk, and

always using a non-cooperative approach. In particular, Lai et al. (2015) evaluate flood

risk in the Dongjiang river basin (China), and Brown and Bhat (2018) focuses on South

Florida’s precipitation trends. In a more general setting, Machac et al. (2017) study flood

risk management for two-player non-cooperative games.

In this article we have selected a sharing rule function to help the planner to distribute

the total benefit among the owners having into account a principles of stability. As far

as we know, our paper presents the first cooperative game theory model applied to flood

risk management.

2 The model

2.1 Cooperative games

A cooperative game is a pair (N, v) where N is a finite set of agents (or players) and

v : 2N −→ R is the characteristic function of the game, where v(S) represents the worth

of coalition S ⊆ N . The interpretation is that the worth of S is the benefit that agents in

S can generate by themselves, without the help of the other agents. As usual, we assume

v(∅) = 0.

A cooperative game (N, v) is superadditive if v(S ∪T ) ≥ v(S)+ v(T ) for all S, T ⊂ N

with S ∩ T = ∅. The interpretation is that two different coalitions can obtain at least as

much benefit working together than by themselves. A cooperative game (N, v) is mono-

tonic if v(S) ≤ v(T ) for all S ⊆ T ⊆ N . The interpretation is that no coalition can obtain

less by adding new members. A cooperative game (N, v) is additive if v(S) =∑

i∈S v(i)

for all S ⊆ N . The interpretation is that there exists no benefit of cooperation, since no

coalition can improve what their members can achieve by themselves alone.

A main objective of cooperative game theory is to select for every cooperative game

an allocation, or a set of allocations, admissible for the players. At this point, two main

approaches are possible. One of them is based on stability, where the aim to find stable

allocations, in the sense that no coalition of players can improve by themselves. The

second one is based on fairness, and it aims to find fair allocations based on some idea of

justice.

5

Let (N, v) be a cooperative game. An imputation of (N, v) is an allocation x ∈ RN

satisfying∑

i∈N xi = v(N) (i.e. the worth of the whole coalition is fully allocated among

its members), and xi ≥ v(i) for all i ∈ N (i.e. no agent gets less than she would get

by herself). We denote as I(N, v) the set of imputations of (N, v). The core of (N, v) is

the set of stable imputations, and it is defined as:

Core(N, v) =

x ∈ I(N, v) :∑

i∈N

xi ≥ v(S) for all S ⊂ N

. (1)

The interpretation of the core is intuitive: We look for payoff allocations that no coalition

of agents can improve by themselves. The main problem with the core is that it may

be empty, as we check in Example 2.3 below. Another (minor) problem is that the core

may be huge, which makes it necessary to find some criteria to pick up a core allocation.

However, if (N, v) is an additive game, we avoid both problems, since the core is a

singleton given by Core(N, v) = x where xi = v(i) for all i ∈ N .

A sharing rule is a function that assigns to each cooperative game (N, v) in some

class of games, a vector φ(N, v) ∈ RN such that

∑

i∈N φi(N, v) = v(N). The most known

sharing rule in cooperative game theory is the Shapley value (Shapley, 1953). In order

to define it formally, we introduce the following notation: Given a finite set N , let ΠN

denote the set of all orders in N . Given π ∈ ΠN , let Pre(i, π) denote the set of elements

of N which come before i in the order given by π, i.e.,

Pre(i, π) = j ∈ N |π(j) < π(i) .

The Shapley value of the cooperative game (N, v) is defined as:

Shi(N, v) =1

n!

∑

π∈ΠN

[v(Pre(i, π) ∪ i)− v(Pre(i, π))]

for all i ∈ N .

2.2 Landflood games

Assume we have a finite number of land owners on a river basin. These will be our agents,

i.e. N is the set of land owners. We denote these agents as N = 1, . . . , n. On the other

hand, we assume that agents are only affected by actions taken by upstream owners,

considering the natural direction of run-off. It refers to the amount of water coming from

rainfall running over the land surface or through the soil to groundwater and streamflow.

Hence, upstream/downstream is defined by a directed graph G with no cycles, whose

nodes are the agents. In particular, (i, j) ∈ G is interpreted as that agent i is upstream

agent j, so that water fallen on region i eventually ends up on agent j’s land.

6

Forest decreases risk of flooding in downstream lands, and they themselves are also

less affected by flooding (Laurance, 2007; Van Dijk et al., 2009) because water retention

potential tends to increase along with the extent of forest cover in a water basin (Tyszka,

2009; European Environment Agency, 2015) and they themselves are also less affected by

flooding due to its retention capacity and water regulation (Licata et al., 2008; Chang,

2012). In general, land owners can use their land for either forests or for other uses.

Woodland covers 165 million hectares in the European Union in 2015, representing 38%

of the territory (Eurostat, 2017). For example, the Rhine Atlas has six different land

uses (te Linde et al., 2011) and in Spain there are thirteen different land uses (SIOSE,

2011), including different types of crops, pastures, scrub, land without vegetation, forest

areas, etc. Taking into account the objective of this study, and in order to keep the model

simple, we regroup the uses in two types: Forests that retain floods and other uses that

accelerate them. For this particular study, we define “forest” as a large tract of land

covered with trees and underbrush (woodland). While term “other uses” includes the

rest of coverages, natural or artificial, that are characterized by the absence of vegetation

and trees.

The profit of having a forest is given by a vector f ∈ RN+ , and the profit given by

other uses that accelerate floods in given by a vector a ∈ RN+ , i.e. when agent i ∈ N has

a forest in her land, she obtains fi ∈ R, and otherwise she obtains ai ∈ R.

Moreover, the positive externality for land j ∈ N due to the presence of a forest in

land i ∈ N is given by a matrix B = (bij)i,j∈N such that bij > 0 when (i, j) ∈ G and

bij = 0 otherwise (“water goes downstream”).

A landflood problem is a tuple (N,G, f, a, B) with the properties given above.

Finally, the expected damage gradually increases downstream (Petts and Amoros,

1996; Graf, 1998; Begum et al., 2007; Llobet et al., 2018). This has been demonstrated

by studies in specific river basins such as te Linde et al. (2011) in the Rhine basin

and Papathanasiou et al. (2013) in the basin of the Ardas river. These areas are the

floodplains, characterised by not having slopes and for being the final evacuation of all

the water that infiltrates in the hydrographic basin, concentrating the highest water flows.

In addition, the larger the forest cover, the more water is retained (Petts and Amoros,

1996; Tyszka, 2009; European Environment Agency, 2015). This again lowers the amount

of water flowing as surface run-off and at the outlets of the catchments. Therefore and

for both reasons, we assume the further the forest, the larger its beneficial effect.

The simplest way to include this assumption in the model is the following:

Assumption 1 (i, j), (j, k) ∈ G implies (i, k) ∈ G and bik ≥ bjk.

7

A landflood game is a cooperative game (N, v) generated by a landflood problem,

where the worth of a coalition S is given by the maximization problem:

v(S) = maxF⊆S

Ψ(F, S, f, a, B)

where1

Ψ(F, S, f, a, B) =

∑

i∈F

fi +∑

j∈S\F

aj +∑

i∈F,j∈S\F

bij

.

In particular, we say that any set in argmaxF⊆N Ψ(F, S, f, a, B) is an optimal config-

uration for S.

Notice that∑

i∈F fi is the profit for having the forests,∑

j∈S\F aj is the profit for

having other uses, and∑

i∈F,j∈S\F bij is the profit due to externalities.

Example 2.1 Let N = 1, 2, 3, f = (1, 0.99, 2) and a = (2, 1, 1). Moreover, the graph

is given by G = (1, 2), (2, 3), (1, 3), i.e. player 1 is upstream, player 3 is downstream,

and player 2 is between both of them (see Figure 1). We assume that the benefit of lands

2 and 3 increase by 2 each when land 1 is a forest, and the benefit of land 3 increases by

1 if land 2 is a forest. Hence, b12 = b13 = 2, b23 = 1, and bij = 0 otherwise.

1

2

32

21

Figure 1: Example of river basin.

The worth of each coalition, as well as the optimal configuration that produces it, is

given in the following table:

S v(S) forest other uses

1 2 - 1

2 1 - 2

3 2 3 -

1, 2 4 1 2

1, 3 4 1 3

2, 3 2.99 2 3

N 7 1 2, 31We use the convention of using i for a generic element F (i.e. agents with forests) and j for a generic

element of N \F (i.e. agents with other uses). When it is undefined whether an agent is a forest or not,

we use either term indistinguishably.

8

Notice that the landflood game given in Example 2.1 satisfies Assumption 1, because

b13 > b23, i.e. for agent 3 it is more favorable to have agent 1 as forest alone than agent

2 as forest alone. In this example, the core is nonempty, as for instance the Shapley

value Sh(N, v) = (2.83, 1.83, 2.33) ∈ Core(N, v). This allocation is achieved by a two-

step procedure: Firstly, optimal configuration F = 1 (i.e. only agent 1 is a forest) is

implemented, so that the direct benefit is (1, 3, 3). Secondly, in compensation for agent 1

being a forest, agent 2 transfers 1.17 units of utility and agent 3 transfers 0.67 of utility

to agent 1.

In order to emphasize the advantage of considering a centralized model as the one we

are proposing in this paper, we briefly compare it with the situation in which the agents

act in a non cooperative way, i.e., without compensations among themselves.

We represent the problem given in Example 2.1 as follows. Assume that we have two

2× 2 matrices so that agent 1 chooses the row, agent 2 chooses the column, and agent 3

chooses the matrix.

3 is a forest

2 forest 2 other uses

1 forest (1, 0.99, 2) (1, 3, 2)

1 other uses (2, 0.99, 2) (2, 1, 2)

3 other uses

2 forest 2 other uses

1 forest (1, 0.99, 4) (1, 3, 3)

1 other uses (2, 0.99, 2) (2, 1, 1)

In order to compute the final payoff allocation in this example, we describe the final

payoff allocation as follows: The first component of each vector is the payment to agent

1, the second component is the payment to agent 2, and the third component is the

payment to agent 3. Each agent has two possible strategies: or devote her land to have

a forest or devote it to other uses.

In this situation, we have the so called iterated elimination of strictly dominated

strategies (Aumann, 1976), which allows us to predict what would be the final outcome

assuming some mild rationality in the agents.

Firstly, agent 1 should choose the second file, since it would give her a larger final

payoff (2) than choosing the first file (1), independently of whatever the other agents do.

Knowing that, agent 2 should choose the second column, since it would give her a larger

final payoff (1) than choosing the first column (0.99), independently of whatever agent

3 does. Knowing that, agent 3 should choose the first matrix, since it would give her a

larger final payoff (2) than choosing the second matrix (1).

Then, the only rational choices in this game are that agents 1 and 2 devote their land

to other uses and agent 3 will be a forest, resulting in a final payoff allocation of (2, 1, 2),

9

which implies that each agent is worse off than with the Shapley value (2.83, 1.83, 2.33).

Not always the Shapley value belongs to the core, as next example shows:

Example 2.2 Let N = 1, 2, 3, 4, 5, f = (0, 0, 0, 0, 1), a = (1, 0, 0, 0, 0). Moreover, the

graph (see Figure 2) is given by G = 〈(1, 2), (1, 3), (1, 4), (1, 5), (2, 5), (3, 5)〉, and bij = 1

for all (i, j) ∈ G.

1

2

3

4

5

11

11

11

Figure 2: Example of landflood game with the Shapley value out of the core.

There exist multiple optimal configurations, as for example F = 1, 5 and also

F ′ = 1, 2, 3. Let (N, v) be the cooperative game generated by this landflood game.

In this case, Sh(N, v) = (1.5, 0.5, 0.5, 0.42, 1.08). However, v(1, 2, 3, 4) = 3 > 2.92 =∑4

i=1 Shi(N, v). Hence, Sh(N, v) /∈ Core(N, v). Nonetheless, Core(N, v) is nonempty,

as for example (1, 0, 1, 1, 1) ∈ Core(N, v).

In next Sections, we propose a method to find core allocations.

Given the private ownership of the land use, any allocation that does not belong to

the core can be blocked by a group of agents, leading to potential loss of efficiency in the

location of the forests.

In general, the assumption “the further the forest, the larger its beneficial effect” is

key for the emptiness of the core, as next example (which does not satisfy Assumption

1) shows:

Example 2.3 Let N = 1, 2, 3, 4, 5, f = (0, 0, 0, 0, 1), a = (1, 0, 0, 0, 0). Moreover, the

graph (see Figure 3) is given by G = 〈(1, 2), (1, 3), (2, 5), (3, 4), (4, 5)〉, b12 = b13 = b25 =

b34 = b45 = 1 and bij = 0 otherwise.

1

2

3 4 5

1

1

1

1 1

Figure 3: Example of graph that induces a cooperative game with empty core.

10

The worth of some coalitions, as well as their respective optimal configurations2, is

given in the following table:

S v(S) forest other uses

1, 2, 3, 4 2 3 1, 2, 4

1, 2, 3, 5 3 1, 5 2, 3

1, 2, 4, 5 3 2, 4 1, 5

1, 3, 4, 5 3 3, 5 1, 4

2, 3, 4, 5 2 3, 5 2, 4

N 3 1, 5 2, 3.

Notice that a core element x in the landflood game given in example 2.3 should satisfy

x1 + x2 + x3 + x4 ≥ 2, x2 + x3 + x4 + x5 ≥ 2, and xi + xj + xk + xl ≥ 3 otherwise, which

implies that x1 + x2 + x3 + x4 + x5 ≥ 3.25. This is not possible because v(N) = 3, and

hence the core is empty for this game.

3 Saturated landflood games

In order to analyze the nonemptyness of the core in general landflood games, we use the

concept of saturated landflood games, defined as follows:

Definition 3.1 We say that a landflood game is saturated if the two following conditions

hold:

• For each pair of adjacent lands, there exists an optimal configuration in which both

of them are forests.

• For each pair of adjacent lands, there exists an optimal configuration in which none

of them are forests.

It is not difficult to check that the landflood problem given in Example 2.3 is saturated.

As opposed, the landflood problem (N, v) presented in Example 2.1 is not saturated, since

the only optimal configuration is F = 1. However, there exits a saturated landflood

game (N,w) satisfying v(S) ≤ w(S) for all S ⊂ N and v(N) = w(N). In view of (1),

this implies that Core(N,w) ⊆ Core(N, v). Hence, the nonemptyness of Core(N,w)

implies the nonemptyness of Core(N, v), and any core allocation in (N,w) is also a core

allocation in (N, v). Next, we show how to generate a possible (N,w) from (N, v) in

Example 2.1. We follow the next steps:

2For S = N , it is irrelevant whether agent 4 is a forest or not.

11

1. Reduce b12 from 2 to 0.99 and increase f1 from 1 to 2.01. With these changes,

v(1) and v(1, 3) increase, whereas the rest of v(S) (including v(N)) remain

unchanged. Furthermore, F = 1, 2 becomes a new optimal configuration.

2. Reduce b23 from 1 to 0, removing arc (3, 4); and increase f2 from 0.99 to 1.99. With

these changes, v(2) and v(2, 3) increase, whereas the rest remain unchanged.

Moreover, agent 1 and 3 become adjacent.

3. Reduce b13 from 2 to 1, and increase f1 from 2.01 to 3.01. With these changes,

v(1) and v(1, 2) increase, whereas the rest remain unchanged. Furthermore, N

and 1, 3 become two new optimal configurations.

4. Reduce b12 from 0.99 to 0, removing arc (1, 2); and increase a2 from 1 to 1.99. With

these changes, each v(S) remains the same.

5. Reduce b13 from 1 to 0, removing arc (1, 3), and increase a3 from 1 to 2. With these

changes, the landflood problem becomes trivially saturated (because there are no

adjacent nodes).

Let (N,w) be resulting landflood game. Then, (N,w) is both saturated and additive

(since there are no externalities). In particular, Core(N,w) = (3.01, 1.99, 2). We then

deduce that (3.01, 1.99, 2) ∈ Core(N, v).

In general, we can replicate this procedure in order to generate a saturated landflood

game from each non-saturated one, as next proposition shows:

Proposition 3.1 For each landflood game (N, v), there exists a saturated landflood game

(N,w) with at most as many arcs and such that

• v(S) ≤ w(S) for all S ⊂ N .

• v(N) = w(N).

Moreover, if (N, v) satisfies Assumption 1, it is possible to find such a (N,w) satisfying

Assumption 1 also.

Proof. Let (N, v) let be a landflood game defined by f , a, and B. We proceed by double

induction on the cardinality of G, |G|, and the cardinality of

Ω =

(i, j) ∈ G : i, j adjacent and maxF⊆N :|F∩i,j|6=1

Ψ(F,N, f, a, B) < v(N)

,

the set of adjacent nodes for which no optimal configuration has either both or none of

them as forests. If G = ∅, then there are no externalities and (N, v) is saturated, so we

12

take w = v. Assume then the result holds when the cardinality of the graph is |G| − 1

or lower. If Ω = ∅, then (N, v) is saturated, and we take w = v. Assume now Ω 6= ∅.

We can assume w.l.o.g. that (1, 2) ∈ Ω, so that 1 is before 2 in the graph. Assume that

no optimal configuration has both 1 and 2 as forests (the case in which none of them are

forest is analogous). We define (N, v′) as follows. Let

F ′ ∈ arg maxF⊆N :1,2⊆F

Ψ(F,N, f, a, B)

be the a configuration with maximum value among those in which both 1 and 2 are forests.

By assumption, this configuration is not optimal, which implies that there exists some

other optimal configuration F ′′ ⊂ N with α = Ψ(F ′′, N, f, a, B) − Ψ(F ′, N, f, a, B) > 0.

Since 1 and 2 are adjacent with 1 before 2 in the graph, we deduce b12 > 0. Hence,

α′ = min α, b12 > 0. Now, define (N, v′) by taking f ′1 = f1 + α′, b′12 = b12 − α′, and

b′i = fi, a′j = aj, and b′ij = bij otherwise. Since, 1 and 2 are adjacent, we deduce that

(N, v′) satisfies Assumption 1 when (N, v) does. It is straightforward to check that F ′′

is still optimal in (N, v′), and so v′(N) = v(N). Moreover, for each S ⊆ N , we also

have v(S) ≤ v′(S), with strict inequality when 1, 2 ∈ S and there exists an optimal

configuration in S in which both 1 and 2 are forests. We have two cases:

1. When α′ = bij, agents 1 and 2 are not adjacent in (N, v′).

2. When α′ = α, F is optimal in (N, v′).

In the first case, we apply the induction hypothesis on |G|. In the second case, we apply

the induction hypothesis on |Ω|. In either case, by the induction hypothesis we know that

there exists a saturated landflood game (N,w) satisfying Assumption 1 if (N, v) does,

and such that v′(S) ≤ w(S) for all S ⊂ N and v′(N) = w(N). Since v(S) ≤ v′(S) for all

S ⊂ N and v′(N) = v(N), we deduce our result.

The relevance of Proposition 3.1 is that Core(N,w) ⊆ Core(N, v), and hence it is

enough to study the nonemptyness of the core for saturated landflood games. Obviously,

Assumption 1 plays a role in this study, since by Example 2.3 we know that there exist

saturated landflood games with empty core when Assumption 1 does not hold.

In Example 2.1 the resulting saturated game has no externalities and hence it is

trivially additive, which allows us to identify a core element. In general, this is not the

case as next example shows:

Example 3.1 Let N = 1, 2, 3, f = (0, 0, 1) and a = (1, 0, 0). Moreover, the graph is

given by G = (1, 2), (2, 3), (1, 3), as in Example 2.1 (see Figure 1). Let b12 = b13 =

b23 = 1.

13

The worth of each coalition and the optimal configurations that generate them are

given in the following table:

S v(S) optimal configurations

1 1 ∅

2 0 ∅, 2

3 1 3

1, 2 1 ∅, 1, 2

1, 3 2 3

2, 3 1 2, 3, 2, 3

N 2 1, 2, 3, 1, 2, 1, 3, 2, 3

It is clear from the table that this landflood problem is saturated.

The landflood problem presented in Example 3.1 is saturated. Moreover, it has exter-

nalities that cannot be reduced by increasing a or f , because it would imply an increase

in v(N). However, it is still possible to remove some adjacent arc (in this case, either

(1, 2) or (2, 3)) without changing the associated landflood game. We claim that this is

true in general:

Claim 3.1 Under Assumption 1, each saturated landflood problem (N,G, f, a, B) satisfies

one of the following conditions:

1. G = ∅, or

2. there exists some (i, j) ∈ G such that v(S) = v−ij(S) for all S ⊆ N , where (N, v) is

the landflood game generated by (N,G, f, a, B) and (N, v−ij) is the landflood game

generated by (N,G \ (i, j), f, a, B′) with b′kl = bkl for all (k, l) ∈ G \ (i, j).

Even though we do not have a formal proof, we have checked it true in more than

640,000 randomly generated landflood games taking natural restrictions. The algorithm

used is described in Appendix.

Claim 3.1 allows us to find core allocations.

Proposition 3.2 Under Claim 3.1, the core is nonempty in any landflood game satisfy-

ing Assumption 1.

Proof. Under Proposition 3.1, for any landflood game (N, v) satisfying Assumption 1,

we can find a saturated landflood game (N,w) with v(S) ≤ w(S) for all S ⊂ N and

14

v(N) = w(N) and satisfying also Assumption 1. Since v(S) ≤ w(S) for all S ⊂ N and

v(N) = w(N), we have Core(N,w) ⊆ Core(N, v) and it is enough to prove Core(N,w) 6=

∅. In case (N,w) has no externalities, it is additive and hence Core(N,w) = x where

xi = w(i) for all i ∈ N . Hence, x ∈ Core(N, v) 6= ∅. In case (N,w) has externalities,

under Claim 3.1.2, we can reduce the cardinality of the graph and repeat the process

until the game becomes additive.

4 Stable sharing rules

Given that we can find core allocations, we look for a way to choose a reasonable one, i.e.

we look for a sharing rule in the class of landflood games. A natural candidate can be

the Shapley value. However, the Shapley value may lay outside the core (Example 2.2),

even when the core is nonempty.

In this section we present three alternative core sharing rules. The first one (Algorithm

1) applies the procedure used in the proofs of Proposition 3.1 and Proposition 3.2 in the

most favorable way for agents located upstream. The second one (Algorithm 2) applies

the algorithm in the most favorable way for agents located downstream. Finally, we

propose an intermediate sharing rule that balances both approaches.

In Algorithm 1, lines 2-10 decrease/remove arcs bij by increasing fi. Once these

transfers are exhausted, lines 11-18 decrease/remove arcs bij by increasing aj. In case of

more than one adjacent arc, the ones that are upstream should be taken first, so that

upstream agents are more favoured. Once these transfers are also exhausted, the problem

is saturated. Line 19 removes arcs unused by any coalition. Under Claim 3.1, we can

always find such arc. In case of more than one, the ones that are downstream should

be taken first, so that upstream agents have the chance to be more favored in the loop.

Once G = ∅, the game is additive and line 20 picks up the stand-alone solution.

In Algorithm 2, preference is given to increase aj before increasing fi. In case of

more than one adjacent arc, the ones that are downstream should be taken first, so that

downstream agents are more favored. Analogously, line 19 removes arcs unused by any

coalition. In case of more than one, the ones that are upstream should be taken first, so

that downstream agents have the chance to be more favored in the loop.

Finally, a compromise value among solutions x and y, provided respectively by Algo-

rithm 1 and Algorithm 2, is the following:

zi =xi + yi

2

for all i ∈ N . Convexity of the core assures that z ∈ Core(N, v).

15

Algorithm 1 Optimal sharing rule for upstream agents

Input: Landflood problem (N,G, f, a, B)

Output: x ∈ RN

1: while G 6= ∅ do

2: for each (i, j) adjacent in G do

3: α← v(N)−maxF⊆N :i,j⊆F Ψ(F,N, f, a, B)

4: if α > 0 then

5: if α ≥ bij then

6: fi ← fi + bij

7: remove arc (i, j) from G

8: else

9: fi ← fi + α

10: bij ← bij − α

11: β ← v(N)−maxF⊂N :F∩i,j=∅ Ψ(F,N, f, a, B)

12: if β > 0 then

13: if β ≥ bij then

14: aj ← aj + bij

15: remove arc (i, j) from G

16: else

17: aj ← aj + β

18: bij ← bij − β

19: remove arc in G unused by any coalition S ⊆ N

20: for each i ∈ N do xi ← maxfi, ai

21: present x as solution.

16

Algorithm 2 Optimal sharing rule for downstream agents

Input: Landflood problem (N,G, f, a, B)

Output: y ∈ RN

1: while G 6= ∅ do

2: for each (i, j) adjacent in G do

3: β ← v(N)−maxF⊂N :F∩i,j=∅ Ψ(F,N, f, a, B)

4: if β > 0 then

5: if β ≥ bij then

6: aj ← aj + bij

7: remove arc (i, j) from G

8: else

9: aj ← aj + β

10: bij ← bij − β

11: α← v(N)−maxF⊆N :i,j⊆F Ψ(F,N, f, a, B)

12: if α > 0 then

13: if α ≥ bij then

14: fi ← fi + bij

15: remove arc (i, j) from G

16: else

17: fi ← fi + α

18: bij ← bij − α

19: remove arc in G unused by any coalition S ⊆ N

20: for each i ∈ N do yi ← maxfi, ai

21: present y as solution.

17

Example 4.1 Following the same procedure as the one previously used to obtain a satu-

rated game in Example 2.1, we compute the three rules for all the examples of the paper.

The obtained results are the following:

x y z

Example 2.1 (3.01, 1.99, 2) (2, 2, 3) (2.505, 1.995, 2.5)

Example 2.2 (1, 1, 1, 0, 1) (1, 1, 1, 0, 1) (1, 1, 1, 0, 1)

Example 3.1 (1, 0, 1) (1, 0, 1) (1, 0, 1)

Discussion

Our study shows the possibility of continuing to progress in the reduction of flood risk

through new alternatives such as incentives to owners to change land use upstream of

the river basin, agreements between management and owners, and allocation of benefits

in case of uses that enhance water retention. This means an advance at a scientific,

technical, political, social and environmental level in this field.

Firstly, the purpose of European Floods Directive (European Commission, 2007) was

to establish a framework for the assessment and management of flood risks, aiming at

the reduction of adverse consequences for human health, environment, cultural heritage

and economic activity associated with floods. With this objective, the Directive requires

Member States to carry out flood risk management plans. Accordingly, public bodies

carry out specific plans to detect the main risks of flooding of the catchments they man-

age, as well as to locate the areas with the highest risk of flooding. The objective of flood

risk management plans is to improve the territory planning and the flood zones man-

agement. In this sense, many studies have tried to model different situations depending

on hydrological and weather data. Some examples are those carried out by Kourgialas

and Karatzas (2011) and Levy (2005), as well as in the review of (Sanyal and Lu, 2004).

Others try to predict future conditions, giving the opportunity to know how will be

the natural response if we modify something. For example, Purvis et al. (2008) offer a

methodology to estimate the probability of future coastal flooding given uncertainty over

possible sea level rise.

On the other hand, the EU Water Framework Directive (European Parliament and

Council, 2000) and the Blueprint to Safeguard Europe’s Water Resources (European

Commission, 2012) also recognise the potential for rural land use change for supporting

water management objectives. In addition, the Directorate-General for Environment

from the European Commission highlights the role of natural approaches for protecting

18

water resources and managing flood risks. It emphases that NWRM are multi-functional

measures that aim to protect and manage water resources and have the potential to

provide multiple benefits, for example: flood risk reduction, water quality improvement,

groundwater recharge, or habitat improvement (European Commission, 2014).

It seems that there are a wide range of different techniques, analyses and studies

that allow us to model situations, make estimates and known risk areas. In addition,

there are different recommendations on techniques of land use management to safeguard

and enhance the water storage potential of landscape, soil, and aquifers. Therefore, it is

recommended to continue advancing on how to avoid these natural risks. It is necessary

to identify synergies to reduce flood risk. Sharing this knowledge in different areas and

sectors is an exercise that should be carried out more frequently than is currently done.

For it, there are other types of tools that are currently not used in the flood risk field but

have great potential. An example is the cooperative game theory that we apply in this

work.

Taking into account all these advances, it seems that a new perspective is possible by

replacing “flood control” with “risk management”. This is possible through influencing

some aspects that can reduce flood risk instead of trying to make a total control. In

addition, this implies social and economic aspects that require the collaboration of all

stakeholders and especially that of the landowners. One of these strategies is to improve

water retention in the territory, controlling, as far as possible, the generation of runoff

that sometimes results in catastrophic floods. It is important to know and simulate the

floods in the cities, because they imply human, economic and environment damages. In

order to do that, it is necessary to consider all the factors that influence the flow of water

that arrive to these cities. These factors are present in all basin, no only downstream.

Therefore, changes, alterations or situations that take place upstream influence what may

happen downstream.

As a result, in this work we evaluate the potential of game theory through cooperative

games as a useful tool in flood risk preventios, as we evaluate what benefits/costs would

cause changes in land use in the upper areas of the catchment.

Conclusions

With the methodology, alternatives and assumptions that are consider in this study, a

more comprehensive and basin-wide approach has been analysed in order to improve

and highlight the importance of retaining water in the catchment and upstream. This

19

has been achieved through the application of cooperative game theory. Moreover, we

explore a barely used tool to solve allocation in flood risk negotiations. Our results

are positive in the sense that they show that stable incentives are possible in order to

encourage landowners to contribute to flood risk reduction. Moreover, our proofs are

constructive. We present two algorithms which actually implement stable compensation

allocations. We can use one or another depending on which kind of land owners (upstream

or downstream) are to be more favoured. An average of both can also be used in order

to implement a more balanced allocation.

Acknowledges

The authors thank Andrea Janeiro, Lars Ehlers and Loe Schlicher, as well as the par-

ticipants at the 14th European Meeting on Game Theory at Bayreuth (Germany), the

XXXVII Congreso Nacional de Estadıstica e Investigacion Operativa at Oviedo (Spain),

and the LAND4FLOOD Workshop and Working Group meeting at Riga (Latvia) for

their useful comments.

Appendix

Algorithm 3 was used to randomly simulate landflood problems. We consider four differ-

ent variables whose values depend on the type of basin that is being analyzed. In this

study, representative values have been considered according to expert criteria in order to

include the widest range of values that can be found in real cases. The main objective

is to carry out a first approach to demonstrate that it is possible to apply a cooperative

game in this kind of conflicts. Other combinations of values are possible.

A) Number of nodes (n = 10 or n = 60). In the first case (n = 10), the objective is to

represent and analyse different cases, among which they may be: small basins, sub-basins,

a small scale of management with special interest (for example in case it is necessary to

make decisions on this scale) and homogeneous basins with the same land use. A greater

number of nodes (n = 60) has been selected to include large basins, more heterogeneous

basins in land use and larger units of land management where a larger scale of analysis

is necessary.

B) Average number of incoming arcs in each node (in = 1, in = 2, or in = 3). This

happens when each node has, at least, one direct upstream neighbour. This may be the

case of large extensions of land. On the other hand, we consider a maximum of three

20

incoming arcs, to represent the other real case of small land owners.

C) Average number of outcoming arcs in each node (out = 1, out = 2, or out = 3).

This is analogous to the incoming arcs case.

D) Ratio between the average benefit/cost of changing from forest to other uses (or

vice-versa) and the average benefit of externalities (r = 0.5 or r = 1). We establish that

the cost / benefit is twice as much upstream than in the node itself (r = 0.5) or, at least,

equivalent (r = 1).

On Algorithm 3, lines 10-12 build H levels in such a way that by randomly creating

out arcs from each node in a level to a respective number of nodes in the next level,

the average number of incoming arcs is in. Lines 13-18 build the network. Lines 19-

20 generate the externalities in such a way that Assumption 1 is satisfied. Lines 21-23

generate vectors f and a in such a way that the expected absolute difference |fi − ai|

divided by∑

(i,j)∈Gbij

|N |is approximately r. Function rand() returns a random number

between 0 and 1.

Algorithm 3 together with Algorithm 1 were implemented using C++ and run on a

64-bit Intel Core i7-4790K CPU 4.00 GHz with 7,7 GiB. In all the instances Claim 3.1

was satisfied. Sample sizes are summarized in the last Table.

21

Algorithm 3 Random landflood problemInput: n, in, out, r

Output: random landflood problem (N,G, f, a, B)

1: if in = out then

2: H ← random integer between 2 and nin.

3: for h = 1, . . . , H do sh ←nH

4: else if in < out then

5: H ← random integer between 2 and log outin

(

1 + n · out−inin2

)

6: s1 ← n ·outin

−1

( outin )

H−1

7: for h = 2, . . . , H do sh ←outin· sh−1

8: else

9: H ← random integer between 2 and log inout

(

1 + n · in−outout2

)

10: sH ← n ·inout

−1

( inout)

H−1

11: for h = H − 1, . . . , 1 do sh ←inout· sh+1

12: adjust s so that sh ∈ N for all h and∑H

h=1 sh = n

13: define S1, . . . , SH partition of N so that |Sh| = sh for all h

14: for h = 1 to H − 1 do

15: for all i ∈ Sh do

16: randomly choose in nodes j1, . . . , jin in Sh+1

17: for l = 1 to in do

18: G ← G ∪ (i, jl) ∪ (j, jl) : (j, i) ∈ G

19: for all (i, j) ∈ G with i ∈ Sl, j ∈ Sm do

20: bij ← random number between m− l − 1 and m− l

21: for all i ∈ N do

22: fi ←3·r|N |·∑

(j,k)∈G bjk · rand()

23: ai ←3·r|N |·∑

(j,k)∈G bjk · rand()

22

Nodesincoming

arcs

outcoming

arcsratio

sample

size

Av. comp.

time (sec.)

10

1

10.5 20,000 0.00018

1 20,000 0.00018

20.5 20,000 0.00006

1 20,000 0.00006

2

10.5 20,000 0.00006

1 20,000 0.00006

20.5 20,000 0.00063

1 20,000 0.00044

30.5 20,000 0.00025

1 20,000 0.00024

3

20.5 20,000 0.00023

1 20,000 0.00023

30.5 20,000 0.00028

1 20,000 0.00029

60

1

10.5 20,000 1.11

1 20,000 2.91

20.5 20,000 0.04

1 20,000 0.03

30.5 20,000 0.06

1 20,000 0.05

2

10.5 20,000 0.02

1 20,000 0.01

20.5 20,000 14.51

1 20,000 7.36

30.5 20,000 2.79

1 >20,000 0.64

3

10.5 20,000 0.02

1 20,000 0.01

20.5 >20,000 0.04

1 20,000 0.10

30.5 20,000 15.09

1 20,000 9.42

TOTAL >640,000

23

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